Parallel Batch-Dynamic Algorithms for Spanners, and Extensions

TL;DR

This paper introduces the first parallel batch-dynamic algorithms for efficiently maintaining graph spanners and sparsifiers with polylogarithmic depth and near-linear amortized work, enabling scalable updates in dynamic graphs.

Contribution

It presents novel parallel algorithms for batch updates in dynamic graphs to maintain spanners and sparsifiers with improved efficiency and scalability.

Findings

Maintains a $(2k-1)$-stretch spanner with near-linear edges.

Creates a sparse spanner with $O(n)$ edges and $ ilde{O}( ext{log} n)$ stretch.

Supports maintaining cut and spectral sparsifiers with $O(n)$ edges.

Abstract

This paper presents the first parallel batch-dynamic algorithms for computing spanners and sparsifiers. Our algorithms process any batch of edge insertions and deletions in an $n$-node undirected graph, in $\text{poly}(\log n)$ depth and using amortized work near-linear in the batch size. Our concrete results are as follows: - Our base algorithm maintains a spanner with $(2k-1)$ stretch and $\tilde{O}(n^{1+1/k})$ edges, for any $k\geq 1$. - Our first extension maintains a sparse spanner with only $O(n)$ edges, and $\tilde{O}(\log n)$ stretch. - Our second extension maintains a $t$-bundle of spanners -- i.e., $t$ spanners, each of which is the spanner of the graph remaining after removing the previous ones -- and allows us to maintain cut/spectral sparsifiers with $\tilde{O}(n)$ edges.